Variable-range hopping

Variable-range hopping is a model used to describe carrier transport in a disordered semiconductor or in amorphous solid by hopping in an extended temperature range. It has a characteristic temperature dependence of Variable-range hopping is a model used to describe carrier transport in a disordered semiconductor or in amorphous solid by hopping in an extended temperature range. It has a characteristic temperature dependence of where β {displaystyle eta } is a parameter dependent on the model under consideration. The Mott variable-range hopping describes low-temperature conduction in strongly disordered systems with localized charge-carrier states and has a characteristic temperature dependence of for three-dimensional conductance (with β {displaystyle eta } = 1/4), and is generalized to d-dimensions Hopping conduction at low temperatures is of great interest because of the savings the semiconductor industry could achieve if they were able to replace single-crystal devices with glass layers. The original Mott paper introduced a simplifying assumption that the hopping energy depends inversely on the cube of the hopping distance (in the three-dimensional case). Later it was shown that this assumption was unnecessary, and this proof is followed here. In the original paper, the hopping probability at a given temperature was seen to depend on two parameters, R the spatial separation of the sites, and W, their energy separation. Apsley and Hughes noted that in a truly amorphous system, these variables are random and independent and so can be combined into a single parameter, the range R {displaystyle extstyle {mathcal {R}}} between two sites, which determines the probability of hopping between them. Mott showed that the probability of hopping between two states of spatial separation R {displaystyle extstyle R} and energy separation W has the form: where α−1 is the attenuation length for a hydrogen-like localised wave-function. This assumes that hopping to a state with a higher energy is the rate limiting process. We now define R = 2 α R + W / k T {displaystyle extstyle {mathcal {R}}=2alpha R+W/kT} , the range between two states, so P ∼ exp ⁡ ( − R ) {displaystyle extstyle Psim exp(-{mathcal {R}})} . The states may be regarded as points in a four-dimensional random array (three spatial coordinates and one energy coordinate), with the 'distance' between them given by the range R {displaystyle extstyle {mathcal {R}}} .